Equations_2_Industrial Instrumentation - Pressure Measurement Important Equations.pdf
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This article provides a summary of all the important equations relating to Pressure Measurement in an industrial environment.
![PRESSURE MEASUREMENT – SUMMARY OF IMPORTANT EQUATIONS
Er. Faruk Bin, Dept. of AEIE, UIT, BU Page 2 of 6
5.
Manometers:
Pressure Balance equation for U – tube manometer
𝑷𝟏 + 𝒈𝒉𝝆𝒇 = 𝑷𝟐 + 𝒈𝒉𝝆𝒎
𝑫𝒊𝒇𝒇𝒆𝒓𝒆𝒏𝒕𝒊𝒂𝒍 𝑷𝒓𝒆𝒔𝒔𝒖𝒓𝒆 (∆𝑷) = 𝑷𝟏 − 𝑷𝟐 = 𝒈𝒉(𝝆𝒎 − 𝝆𝒇) = 𝒈𝒉𝝆𝒎
𝜌𝑚 = 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑚𝑎𝑛𝑜𝑚𝑒𝑡𝑒𝑟 𝑚𝑒𝑟𝑐𝑢𝑟𝑦;
𝜌𝑓 = 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑡𝑟𝑎𝑛𝑠𝑚𝑖𝑡𝑡𝑖𝑛𝑔 𝑓𝑙𝑢𝑖𝑑
General Manometer
∆𝑷 = (𝑷𝟏 − 𝑷𝟐) = (𝝆𝒎 − 𝝆𝒇)(𝒉𝟏 − 𝒉𝟐)𝒈 = (𝝆𝒎 − 𝝆𝒇)𝒉𝒈
U-Tube Manometer
∆𝑷 = (𝑷𝟏 − 𝑷𝟐) = (𝝆𝒎 − 𝝆𝒇)(𝒉𝟏 − 𝒉𝟐)𝒈 = (𝝆𝒎 − 𝝆𝒇)𝒉𝒈
Differential U-Tube Manometer
∆𝑷 = (𝑷𝟏 − 𝑷𝟐) = (𝝆𝒎 − 𝝆𝒇)(𝒉𝟏 − 𝒉𝟐)𝒈 = (𝝆𝒎 − 𝝆𝒇)𝒉𝒈
Inverted U-Tube Manometer
∆𝑷 = 𝑷𝟏 − 𝑷𝟐 = 𝝆𝟏𝒈𝑯𝟏 + 𝝆𝟐𝒈𝑯𝟐 − 𝝆𝒎𝒈(𝑯𝟏 − 𝑯𝟐)
Inclined Tube Manometer
∆𝑷 = 𝑷𝟏 − 𝑷𝟐 = 𝝆𝒈(𝒙 + 𝒚) = 𝝆𝒈𝒅 (
𝑨𝟐
𝑨𝟏
+ 𝒔𝒊𝒏𝜽)
[𝑦 = 𝑑𝑠𝑖𝑛𝜃, 𝐴1𝑥 = 𝐴2𝑑], 𝑑 = 𝑐𝑎𝑝𝑖𝑙𝑙𝑎𝑟𝑦 𝑡𝑢𝑏𝑒 𝑙𝑒𝑛𝑔𝑡ℎ 𝑑𝑟𝑢𝑚 𝑑𝑎𝑡𝑢𝑚 𝑝𝑜𝑖𝑛𝑡.
∆𝑷 = 𝑷𝟏 − 𝑷𝟐 = 𝝆𝒈𝒉′(𝟏 + 𝑨𝟐 𝑨𝟏
⁄ ) = 𝝆𝒈𝑹𝒔𝒊𝒏𝜽(𝟏 + 𝑨𝟐 𝑨𝟏
⁄ )
ℎ′
= 𝑅𝑠𝑖𝑛𝜃, 𝑅 = 𝑆𝑐𝑎𝑙𝑒 𝑟𝑒𝑎𝑑𝑖𝑛𝑔, 𝜃 = 𝑎𝑛𝑔𝑙𝑒 𝑜𝑓 𝑖𝑛𝑐𝑙𝑖𝑛𝑎𝑡𝑖𝑜𝑛
Micro Manometer
∆𝑷 = 𝑷𝟐 − 𝑷𝟏 = 𝝆𝒈(𝒉𝟐 − 𝒉𝟏) = 𝝆𝒈(𝒍𝒔𝒊𝒏𝜽 − 𝒉𝟏)
Well-Type Manometer
∆𝑷 = 𝑷𝟏 − 𝑷𝟐 = 𝝆𝒈𝒉 (𝟏 +
𝑨𝟐
𝑨𝟏
) = 𝝆𝒈𝒉
[𝑖𝑓 𝐴2 𝐴1
⁄ 𝑣𝑎𝑙𝑢𝑒 𝑖𝑠 𝑡𝑜𝑜 𝑠𝑚𝑎𝑙𝑙]
𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 𝑖𝑛 𝑤𝑒𝑙𝑙 (𝑑) = ℎ − ℎ′
𝑑𝐴1 = ℎ𝐴2](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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Equations_2_Industrial Instrumentation - Pressure Measurement Important Equations.pdf
- 1. PRESSURE MEASUREMENT – SUMMARY OF IMPORTANT EQUATIONS Er. Faruk Bin, Dept. of AEIE, UIT, BU Page 1 of 6 Industrial Instrumentation Pressure Measurement Sl. No. 1. Pressure 𝑷𝒓𝒆𝒔𝒔𝒖𝒓𝒆 (𝑷) = 𝑭𝒐𝒓𝒄𝒆 𝑨𝒓𝒆𝒂 = 𝑭 𝑨 Pressure for an ideal gas is expressed as 𝒑 = 𝟏 𝟑 𝒎𝒏𝒗𝒓𝒎𝒔 𝟐 𝑛 = 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑎𝑟 𝑑𝑒𝑛𝑠𝑖𝑡𝑦; 𝑚 = 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑎𝑟 𝑚𝑎𝑠𝑠; 𝑣𝑟𝑚𝑠 = 𝑟𝑚𝑠 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑜𝑓 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒 Pressure Sensitivity (S) 𝑺 = 𝑶𝒖𝒕𝒑𝒖𝒕 𝑰𝒏𝒑𝒖𝒕 = ∆𝒉 ∆𝒑 = 𝒉 𝒈𝒉𝝆𝒎 = 𝟏 𝒈𝝆𝒎 2. Pascal – SI Unit for Pressure 𝟏 𝑷𝒂 = 𝟏 𝑵𝒆𝒘𝒕𝒐𝒏 𝒇𝒐𝒓𝒄𝒆 𝒂𝒑𝒑𝒍𝒊𝒆𝒅 𝒑𝒆𝒓 𝒔𝒒𝒖𝒂𝒓𝒆 𝒎𝒆𝒕𝒆𝒓 = 𝟏 𝑵 𝟏 𝒎𝟐 𝑷𝒂 = 𝑵 𝒎𝟐 ⁄ 3. Absolute Pressure 𝑨𝒃𝒔𝒐𝒍𝒖𝒕𝒆 𝑷𝒓𝒆𝒔𝒔𝒖𝒓𝒆 = 𝑮𝒂𝒖𝒈𝒆 𝑷𝒓𝒆𝒔𝒔𝒖𝒓𝒆 + 𝑨𝒕𝒎𝒐𝒔𝒑𝒉𝒆𝒓𝒊𝒄 𝑷𝒓𝒆𝒔𝒔𝒖𝒓𝒆 Relationship between Absolute 𝒑𝒂, Gauge 𝒑𝒈, Atmospheric 𝒑𝒔and Vacuum Pressure 𝒑𝒗 𝒑𝒈 = 𝒑𝒂 − 𝒑𝒔 𝒑𝒗 = 𝒑𝒔 − 𝒑𝒂 𝑽𝒆𝒍𝒐𝒄𝒊𝒕𝒚 𝑷𝒓𝒆𝒔𝒔𝒖𝒓𝒆 = 𝑻𝒐𝒕𝒂𝒍 𝑷𝒓𝒆𝒔𝒔𝒖𝒓𝒆 − 𝑺𝒕𝒂𝒕𝒊𝒄 𝑷𝒓𝒆𝒔𝒔𝒖𝒓𝒆 4. Units 𝟏 𝒂𝒕𝒎 = 𝟏𝟒. 𝟔𝟗𝟔 𝒑𝒔𝒊 = 𝟏𝟎𝟏. 𝟑𝟐𝟓 𝒌𝑷𝒂 𝟏 𝒎𝒊𝒍𝒍𝒊𝒃𝒂𝒓 = 𝟏𝟎𝟎 𝒅𝒚𝒏𝒆 𝒄𝒎𝟐 ⁄ = 𝟏𝟒. 𝟓 ∗ 𝟏𝟎−𝟑 𝒑𝒔𝒊 𝟏 𝒎𝒊𝒄𝒓𝒐𝒏 = 𝟏𝟎−𝟔 𝒎 𝒐𝒓𝟏𝟎−𝟑 𝒎𝒎 𝑯𝒈 = 𝟏𝟗. 𝟑𝟒 ∗ 𝟏𝟎−𝟔 𝒑𝒔𝒊 𝟏 𝑩𝒂𝒓 = 𝟏𝟎𝟓 𝑷𝒂 𝟏 𝒂𝒕𝒎 = 𝟕𝟔 𝒄𝒎 𝒐𝒇 𝑯𝒈 (𝑴𝒆𝒓𝒄𝒖𝒓𝒚) = 𝟏𝟒. 𝟕 𝒑𝒔𝒊 = 𝟏𝟎𝟏. 𝟑 𝒌𝑷𝒂; 𝟏𝒄𝒎 𝒐𝒇 𝑯𝒈 = 𝟏𝟑𝟑𝟑. 𝟐𝟐 𝑷𝒂; 𝟏 𝑷𝒂 = 𝟎. 𝟎𝟎𝟎𝟕𝟓 𝒄𝒎 𝒐𝒇 𝑯𝒈
- 2. PRESSURE MEASUREMENT – SUMMARY OF IMPORTANT EQUATIONS Er. Faruk Bin, Dept. of AEIE, UIT, BU Page 2 of 6 5. Manometers: Pressure Balance equation for U – tube manometer 𝑷𝟏 + 𝒈𝒉𝝆𝒇 = 𝑷𝟐 + 𝒈𝒉𝝆𝒎 𝑫𝒊𝒇𝒇𝒆𝒓𝒆𝒏𝒕𝒊𝒂𝒍 𝑷𝒓𝒆𝒔𝒔𝒖𝒓𝒆 (∆𝑷) = 𝑷𝟏 − 𝑷𝟐 = 𝒈𝒉(𝝆𝒎 − 𝝆𝒇) = 𝒈𝒉𝝆𝒎 𝜌𝑚 = 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑚𝑎𝑛𝑜𝑚𝑒𝑡𝑒𝑟 𝑚𝑒𝑟𝑐𝑢𝑟𝑦; 𝜌𝑓 = 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑡𝑟𝑎𝑛𝑠𝑚𝑖𝑡𝑡𝑖𝑛𝑔 𝑓𝑙𝑢𝑖𝑑 General Manometer ∆𝑷 = (𝑷𝟏 − 𝑷𝟐) = (𝝆𝒎 − 𝝆𝒇)(𝒉𝟏 − 𝒉𝟐)𝒈 = (𝝆𝒎 − 𝝆𝒇)𝒉𝒈 U-Tube Manometer ∆𝑷 = (𝑷𝟏 − 𝑷𝟐) = (𝝆𝒎 − 𝝆𝒇)(𝒉𝟏 − 𝒉𝟐)𝒈 = (𝝆𝒎 − 𝝆𝒇)𝒉𝒈 Differential U-Tube Manometer ∆𝑷 = (𝑷𝟏 − 𝑷𝟐) = (𝝆𝒎 − 𝝆𝒇)(𝒉𝟏 − 𝒉𝟐)𝒈 = (𝝆𝒎 − 𝝆𝒇)𝒉𝒈 Inverted U-Tube Manometer ∆𝑷 = 𝑷𝟏 − 𝑷𝟐 = 𝝆𝟏𝒈𝑯𝟏 + 𝝆𝟐𝒈𝑯𝟐 − 𝝆𝒎𝒈(𝑯𝟏 − 𝑯𝟐) Inclined Tube Manometer ∆𝑷 = 𝑷𝟏 − 𝑷𝟐 = 𝝆𝒈(𝒙 + 𝒚) = 𝝆𝒈𝒅 ( 𝑨𝟐 𝑨𝟏 + 𝒔𝒊𝒏𝜽) [𝑦 = 𝑑𝑠𝑖𝑛𝜃, 𝐴1𝑥 = 𝐴2𝑑], 𝑑 = 𝑐𝑎𝑝𝑖𝑙𝑙𝑎𝑟𝑦 𝑡𝑢𝑏𝑒 𝑙𝑒𝑛𝑔𝑡ℎ 𝑑𝑟𝑢𝑚 𝑑𝑎𝑡𝑢𝑚 𝑝𝑜𝑖𝑛𝑡. ∆𝑷 = 𝑷𝟏 − 𝑷𝟐 = 𝝆𝒈𝒉′(𝟏 + 𝑨𝟐 𝑨𝟏 ⁄ ) = 𝝆𝒈𝑹𝒔𝒊𝒏𝜽(𝟏 + 𝑨𝟐 𝑨𝟏 ⁄ ) ℎ′ = 𝑅𝑠𝑖𝑛𝜃, 𝑅 = 𝑆𝑐𝑎𝑙𝑒 𝑟𝑒𝑎𝑑𝑖𝑛𝑔, 𝜃 = 𝑎𝑛𝑔𝑙𝑒 𝑜𝑓 𝑖𝑛𝑐𝑙𝑖𝑛𝑎𝑡𝑖𝑜𝑛 Micro Manometer ∆𝑷 = 𝑷𝟐 − 𝑷𝟏 = 𝝆𝒈(𝒉𝟐 − 𝒉𝟏) = 𝝆𝒈(𝒍𝒔𝒊𝒏𝜽 − 𝒉𝟏) Well-Type Manometer ∆𝑷 = 𝑷𝟏 − 𝑷𝟐 = 𝝆𝒈𝒉 (𝟏 + 𝑨𝟐 𝑨𝟏 ) = 𝝆𝒈𝒉 [𝑖𝑓 𝐴2 𝐴1 ⁄ 𝑣𝑎𝑙𝑢𝑒 𝑖𝑠 𝑡𝑜𝑜 𝑠𝑚𝑎𝑙𝑙] 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 𝑖𝑛 𝑤𝑒𝑙𝑙 (𝑑) = ℎ − ℎ′ 𝑑𝐴1 = ℎ𝐴2
- 3. PRESSURE MEASUREMENT – SUMMARY OF IMPORTANT EQUATIONS Er. Faruk Bin, Dept. of AEIE, UIT, BU Page 3 of 6 𝒉 = 𝒉′ (𝟏 + 𝑨𝟐 𝑨𝟏 ) 𝐴2 = 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑢𝑏𝑒; 𝐴1 = 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑤𝑒𝑙𝑙; ℎ = 𝑚𝑒𝑟𝑐𝑢𝑟𝑦 𝑐𝑜𝑙𝑢𝑚𝑛 ℎ𝑒𝑖𝑔ℎ𝑡 Ring-Balance Manometer ∆𝑷 = 𝑷𝟏 − 𝑷𝟐 = 𝑮 ∗ 𝑺 𝑨 ∗ 𝒓 𝐬𝐢𝐧 𝜶 𝛼 = 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑙 𝑎𝑛𝑔𝑙𝑒, 𝑟 = 𝑡𝑢𝑏𝑒 𝑟𝑎𝑑𝑖𝑢𝑠, 𝐴 = 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑙 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛, 𝐺 = 𝑏𝑎𝑙𝑎𝑛𝑐𝑖𝑛𝑔 𝑤𝑒𝑖𝑔ℎ𝑡, 𝑆 = 𝑡𝑢𝑏𝑖𝑛𝑔 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒. Barometer 𝑷𝒂𝒕𝒎 = 𝝆𝑯𝒈𝒈𝒉 𝑷𝑺 = 𝑷𝑺𝟎 (𝟏 − 𝑩𝒁 𝑻𝟎 ) 𝟓.𝟐𝟔 𝑃𝑆 = 𝑙𝑜𝑐𝑎𝑙 𝑎𝑡𝑚𝑜𝑠𝑝ℎ𝑒𝑟𝑖𝑐 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒, 𝑃𝑆0 = 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑎𝑡𝑚𝑜𝑠𝑝ℎ𝑒𝑟𝑖𝑐 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑎𝑡 𝑠𝑒𝑎 𝑙𝑒𝑣𝑒𝑙, 760 𝑚𝑚 𝑜𝑓 𝐻𝑔, 𝑍 = 𝑎𝑙𝑡𝑖𝑡𝑢𝑑𝑒 𝑖𝑛 𝑚𝑒𝑡𝑟𝑒, 𝐵 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 = 0.0065 𝐾 𝑚 ⁄ , 𝑇0 = 𝑎𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 = 273.16 𝐾 6. Manometer Transfer Function 𝑇𝑜𝑡𝑎𝑙 𝐹𝑜𝑟𝑐𝑒 𝑒𝑥𝑒𝑟𝑡𝑒𝑑 𝑜𝑛 𝑡ℎ𝑒 𝑙𝑖𝑞𝑢𝑖𝑑 = 𝜌𝑎𝑔(𝐻 − ℎ) = 2𝜌𝑎𝑔𝑥 𝐹𝑜𝑟𝑐𝑒 𝑎𝑐𝑡𝑖𝑛𝑔 𝑑𝑢𝑒 𝑡𝑜 𝑖𝑛𝑒𝑟𝑡𝑖𝑎 = 𝜌𝑉 𝑑2 𝑥 𝑑𝑡2 𝐹𝑜𝑟𝑐𝑒 𝑎𝑐𝑡𝑖𝑛𝑔 𝑑𝑢𝑒 𝑡𝑜 𝑣𝑖𝑠𝑐𝑜𝑢𝑠 𝑓𝑟𝑖𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑓𝑙𝑢𝑖𝑑 = 𝐵 𝑑𝑥 𝑑𝑡 𝑉 = 𝑇𝑜𝑡𝑎𝑙 𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑚𝑎𝑛𝑜𝑚𝑒𝑡𝑟𝑖𝑐 𝑓𝑙𝑢𝑖𝑑; 𝑎 = 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡𝑢𝑏𝑒; 𝐵 = 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑣𝑖𝑠𝑐𝑜𝑠𝑖𝑡𝑦; 𝑥 = 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑟𝑛𝑡 𝑜𝑓 𝑓𝑙𝑢𝑖𝑑; 𝑝 = 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑎𝑝𝑝𝑙𝑖𝑒𝑑 By balancing the forces, we get 𝑇𝑜𝑡𝑎𝑙 𝐹𝑜𝑟𝑐𝑒 𝑑𝑢𝑒 𝑡𝑜 𝑝 = 𝑇𝑜𝑡𝑎𝑙 𝑓𝑜𝑟𝑐𝑒 𝑜𝑛 𝑙𝑖𝑞𝑢𝑖𝑑 + 𝐹𝑜𝑟𝑐𝑒 𝑑𝑢𝑒 𝑡𝑜 𝑖𝑛𝑒𝑟𝑡𝑖𝑎 + 𝐹𝑜𝑟𝑐𝑒 𝑑𝑢𝑒 𝑡𝑜 𝑣𝑖𝑠𝑐𝑜𝑢𝑠 𝑓𝑟𝑖𝑐𝑡𝑖𝑜𝑛 𝑝𝑎 = 𝜌𝑉 𝑑2 𝑥 𝑑𝑡2 + 𝐵 𝑑𝑥 𝑑𝑡 + 2𝜌𝑔𝑎𝑥 ∴ 𝒑 = 𝝆𝑽 𝒂 𝒅𝟐 𝒙 𝒅𝒕𝟐 + 𝑩 𝒂 𝒅𝒙 𝒅𝒕 + 𝟐𝝆𝒈𝒙 Taking Laplace Transfer on both sides, we get 𝑝(𝑠) = [ 𝜌𝑉 𝑎 𝑠2 + 𝐵 𝑎 𝑠 + 2𝜌𝑔] 𝑋(𝑠)
- 4. PRESSURE MEASUREMENT – SUMMARY OF IMPORTANT EQUATIONS Er. Faruk Bin, Dept. of AEIE, UIT, BU Page 4 of 6 𝑇𝑟𝑎𝑛𝑠𝑓𝑒𝑟 𝐹𝑢𝑛𝑐𝑡𝑖𝑜𝑛 = 𝐺(𝑠) = 𝐿𝑎𝑝𝑙𝑎𝑐𝑒 𝑜𝑓 𝑜𝑢𝑡𝑝𝑢𝑡 𝐿𝑎𝑝𝑙𝑎𝑐𝑒 𝑜𝑓 𝑖𝑛𝑝𝑢𝑡 = 𝑋(𝑠) 𝑃(𝑠) 𝐺(𝑠) = 2𝜌𝑎𝑔. 𝑋(𝑠) 𝑝(𝑠) 𝑮(𝒔) = 𝟐𝒂𝒈 𝑽 ⁄ 𝒔𝟐 + 𝑩 𝝆𝑽 𝒔 + 𝟐𝒂𝒈 𝑽 𝑛𝑎𝑡𝑢𝑟𝑎𝑙 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝜔𝑛 = √2𝑎𝑔 𝑉 ⁄ ; 𝑑𝑎𝑚𝑝𝑖𝑛𝑔 𝑟𝑎𝑡𝑖𝑜𝑛 𝜉 = 𝐵 2𝜌√2𝑎𝑔𝑉 𝑪(𝒔) = 𝝎𝒏 𝟐 𝒔𝟐 + 𝟐𝝃𝝎𝒏𝒔 + 𝝎𝒏 𝟐 For Elastic Pressure Transducer 𝑷 = 𝑷𝒇𝒊𝒏𝒂𝒍 + (𝑷𝒊𝒏𝒊𝒕𝒊𝒂𝒍 − 𝑷𝒇𝒊𝒏𝒂𝒍)𝒆−𝒕 𝝉 ⁄ 𝑃 = 𝑎𝑛𝑦𝑡𝑖𝑚𝑒 𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒, 𝑃𝑓𝑖𝑛𝑎𝑙 = 𝐹𝑖𝑛𝑎𝑙 𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒, 𝑃𝑖𝑛𝑖𝑡𝑖𝑎𝑙 = 𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒, 𝜏 = 𝑡𝑖𝑚𝑒 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡, 𝑡 = 𝑎𝑛𝑦 𝑡𝑖𝑚𝑒 𝑖𝑛𝑠𝑡𝑎𝑛𝑡, 7. Vacuum and Low Pressure McLeod Gauge 𝑉 𝑑𝑝2 𝑑𝑡 = 𝐾(𝑃1 − 𝑃2) 𝑉 = 𝑏𝑢𝑙𝑏 𝑣𝑜𝑙𝑢𝑚𝑒, 𝐾 = 𝑓𝑙𝑜𝑤 𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑎𝑛𝑐𝑒, 𝑑𝑝2 𝑑𝑡 = 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑔𝑟𝑎𝑑𝑖𝑒𝑛𝑡 𝑖𝑛 𝑡𝑖𝑚𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑡𝑤𝑜 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑷 = 𝒉𝟐 𝑨𝒕 𝑽 − 𝒉. 𝑨𝒕 𝑷 = 𝒉𝟐 𝑨𝒕 𝑽 ℎ = ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑒𝑑 𝑓𝑙𝑢𝑖𝑑 𝑐𝑜𝑙𝑢𝑚𝑛; 𝐴𝑡 = 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑐𝑟𝑜𝑠𝑠 𝑠𝑒𝑐𝑡𝑖𝑜𝑛; 𝑉 = 𝑣𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑏𝑢𝑙𝑏 Ionization Gauge 𝑆 = 𝑂𝑢𝑡𝑝𝑢𝑡 𝐼𝑛𝑝𝑢𝑡 = 𝑰𝑷 𝑃𝑰𝑮 𝑷 = 𝟏 𝑺 ∗ 𝑰𝑷 𝑰𝑮 𝑆 𝑖𝑠 𝐺𝑎𝑢𝑔𝑒 𝑆𝑒𝑛𝑠𝑖𝑡𝑖𝑣𝑖𝑡𝑦, 𝐼𝑃 𝑖𝑠 𝑃𝑙𝑎𝑡𝑒 𝐶𝑢𝑟𝑟𝑒𝑛𝑡 & 𝐼𝐺 𝑖𝑠 𝐺𝑟𝑖𝑑 𝐶𝑢𝑟𝑟𝑒𝑛𝑡 Knudsen Gauge 𝑷 = 𝒌𝑭 √𝑻𝑷 𝑻𝒈 ⁄ − 𝟏 = 𝟒𝑭 𝑻𝒈 𝑻𝑷 − 𝑻𝑽
- 5. PRESSURE MEASUREMENT – SUMMARY OF IMPORTANT EQUATIONS Er. Faruk Bin, Dept. of AEIE, UIT, BU Page 5 of 6 𝐹 = 𝑁𝑒𝑡 𝑓𝑜𝑟𝑐𝑒; 𝑘 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 = 4, 𝑇𝑔 = 𝑇𝑒𝑚𝑝. 𝑜𝑓 𝑔𝑎𝑠; 𝑇𝑉 = 𝑣𝑎𝑛𝑒 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒, 𝑇𝑃 = 𝑝𝑙𝑎𝑡𝑒 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 8. High Pressure Bridgman Gauge 𝑹 = 𝑹𝟏(𝟏 + 𝒃∆𝑷) 𝑅 = 𝑤𝑖𝑟𝑒 𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒, 𝑅1 = 𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑎𝑡 1 𝑎𝑡𝑚, 𝑏 = 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒 = 2.5 ∗ 10−11 𝑃𝑎−1 , ∆𝑃 = 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑙 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 9. Elastic Pressure Transducer Bourdon Tube: ∆𝒂 = 𝟎. 𝟎𝟓 𝒂𝑷 𝑬 ( 𝒓 𝒕 ) 𝟎.𝟐 ( 𝒙 𝒚 ) 𝟎.𝟑𝟑 ( 𝒙 𝒕 ) 𝟑 ∆𝑎 = 𝑡𝑖𝑝 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡, 𝑎 = 𝑡𝑖𝑝 𝑚𝑜𝑣𝑒𝑚𝑒𝑛𝑡, 𝐸 = 𝑌𝑜𝑢𝑛𝑔′ 𝑠𝑀𝑜𝑑𝑢𝑙𝑢𝑠, 𝑟 = 𝑑𝑖𝑎𝑙 𝑟𝑎𝑑𝑖𝑢𝑠, 𝑥 = 𝑙𝑒𝑛𝑔𝑡ℎ, 𝑦 = 𝑤𝑖𝑑𝑡ℎ, 𝑡 = 𝑒𝑙𝑒𝑚𝑒𝑛𝑡 𝑡ℎ𝑖𝑐𝑘𝑛𝑒𝑠𝑠, Bellow: 𝒅 = 𝟎. 𝟒𝟓𝟑𝑷. 𝒓. 𝒏. 𝑫𝟐 √𝟏 − 𝝑𝟐 𝑬𝒕𝟑 𝑑 = 𝑏𝑒𝑙𝑙𝑜𝑤 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡, 𝑃 = 𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒, 𝑟 = 𝑟𝑎𝑖𝑠𝑢 𝑜𝑓 𝑒𝑎𝑐ℎ 𝑐𝑜𝑟𝑟𝑢𝑔𝑎𝑡𝑖𝑜𝑛, 𝑛 = 𝑛𝑜. 𝑜𝑓 𝑠𝑒𝑚𝑖 − 𝑐𝑖𝑟𝑐𝑢𝑙𝑎𝑟 𝑐𝑜𝑟𝑟𝑢𝑔𝑎𝑡𝑖𝑜𝑛, 𝑡 = 𝑤𝑎𝑙𝑙 𝑡ℎ𝑖𝑐𝑘𝑛𝑒𝑠𝑠, 𝐷 = 𝑚𝑒𝑎𝑛 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟, 𝐸 = 𝑌𝑜𝑢𝑛𝑔′ 𝑠𝑚𝑜𝑑𝑢𝑙𝑢𝑠, 𝜗 = 𝑃𝑜𝑖𝑠𝑜𝑛′ 𝑠 𝑟𝑎𝑡𝑖𝑜, Diaphragm: 𝒅𝒎 = 𝟑𝑷 𝟏𝟔𝑬𝒕𝟑 𝑹𝟒(𝟏 − 𝝑𝟐) 𝒅𝒓 = 𝟑𝑷(𝟏 − 𝝑𝟐) 𝟏𝟔𝑬𝒕𝟑 (𝑹𝟐 − 𝒓𝟐)𝟐 𝒇𝒏 = 𝟐. 𝟓𝒕 𝝅𝑹𝟐 [ 𝑬 𝟑𝝆 (𝟏 − 𝝑𝟐)] 𝟏 𝟐 ⁄ 𝑑𝑚 = 𝑚𝑎𝑥𝑖𝑚𝑢𝑚 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡, 𝑑𝑟 = 𝑑𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛 𝑎𝑡 𝑟𝑎𝑑𝑖𝑢𝑠, 𝐸 = 𝑌𝑜𝑢𝑛𝑔′ 𝑠 𝑚𝑜𝑑𝑢𝑙𝑢𝑠, 𝑃 = 𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒, 𝑡 = 𝑡ℎ𝑖𝑐𝑘𝑛𝑒𝑠𝑠, 𝜗 = 𝑃𝑜𝑖𝑠𝑠𝑜𝑛′ 𝑠 𝑟𝑎𝑡𝑖𝑜, 𝑅 = 𝑟𝑎𝑑𝑖𝑢𝑠, 𝑟 = 𝑟𝑎𝑑𝑖𝑎𝑙 𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛, 𝑓𝑛 = 𝑛𝑎𝑡𝑢𝑟𝑎𝑙 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦, 𝜌 = 𝑚𝑎𝑡𝑒𝑟𝑖𝑎𝑙 𝑑𝑒𝑛𝑠𝑖𝑡𝑦, 10. Electrical Pressure Transducers Strain Gauge Strain Gauge Resistance (R) 𝑹 = 𝝆 𝒍 𝑨
- 6. PRESSURE MEASUREMENT – SUMMARY OF IMPORTANT EQUATIONS Er. Faruk Bin, Dept. of AEIE, UIT, BU Page 6 of 6 𝑅 = 𝑆𝑡𝑟𝑎𝑖𝑛 𝐺𝑎𝑢𝑔𝑒 𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒, 𝜌 = 𝑟𝑒𝑠𝑖𝑠𝑡𝑖𝑣𝑖𝑡𝑦 = 𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑜𝑓 𝑠𝑡𝑟𝑎𝑖𝑛 𝑔𝑎𝑢𝑔𝑒, 𝑙 = 𝑠𝑡𝑟𝑎𝑖𝑛 𝑔𝑎𝑢𝑔𝑒 𝑙𝑒𝑛𝑔𝑡ℎ, 𝐴 = 𝑐𝑟𝑜𝑠𝑠 − 𝑠𝑒𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑠𝑡𝑟𝑎𝑖𝑛 𝑔𝑎𝑢𝑔𝑒. Strain (𝜺) 𝜺 = 𝟏 𝑲 ∆𝑹 𝑹 𝜀 = 𝑆𝑡𝑟𝑎𝑖𝑛, 𝐾 = 𝑠𝑡𝑟𝑎𝑖𝑛 𝑔𝑎𝑢𝑔𝑒 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡, ∆𝑅 𝑅 ⁄ = 𝑣𝑎𝑟𝑖𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑠𝑡𝑟𝑎𝑖𝑛 𝑔𝑎𝑢𝑔𝑒 𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒 Hooke’s Law for Stress (𝝈) 𝝈 = 𝜺. 𝑬 = 𝟏 𝑲 ∆𝑹 𝑹 . 𝑬 𝜎 = 𝑆𝑡𝑟𝑒𝑠𝑠 (𝑁 𝑚2 ⁄ ), 𝜀 = 𝑠𝑡𝑟𝑎𝑖𝑛(𝑚 𝑚 ⁄ ), 𝐸 = 𝑌𝑜𝑢𝑛𝑔′ 𝑠𝑒𝑙𝑎𝑠𝑡𝑖𝑐𝑖𝑡𝑦 𝑚𝑜𝑑𝑢𝑙𝑢𝑠 Capacitive Pressure Gauge 𝑪 = ∈𝟎∈𝒓 𝑨 𝒅 C = capacitance in Farad; A = area of each plate (𝑚2), 𝑑 = 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑝𝑙𝑎𝑡𝑒𝑠 (𝑚), 𝜖0 = 𝑑𝑖𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 = 8.854 ∗ 10−12 𝐹 𝑚2 ⁄ , 𝜖𝑟 = 𝑑𝑖𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 = 1 𝑓𝑜𝑟 𝑎𝑖𝑟 𝑜𝑟 𝑓𝑟𝑒𝑒 𝑠𝑝𝑎𝑐𝑒